To graph the function $f(x) = [x+2]$, which represents the greatest integer function (or floor function) shifted by 2 units, we can break it into the following steps:

Steps:

1. Base Function: Recall the greatest integer function $f(x) = [x]$, which outputs the largest integer less than or equal to $x$.
2. Horizontal Shift: The function $f(x) = [x+2]$ shifts the graph of $[x]$ to the left by 2 units.
3. Graph Structure: The graph consists of horizontal line segments at integer steps, with open circles at the left end and closed circles at the right end of each interval.

Now, I will create a graph to show these steps. matplotlib.pyplot as plt import numpy as np

Define the function for greatest integer with a shift

def greatest_integer_shift(x): return np.floor(x + 2)

Generate x values for the plot

x = np.linspace(-5, 3, 1000) y = greatest_integer_shift(x)

Plot the function

plt.figure(figsize=(8, 6)) plt.step(x, y, where="post", label=r"$f(x) = \lfloor x+2 \rfloor$") plt.title("Graph of $f(x) = [x+2]$ (Greatest Integer Function)", fontsize=14) plt.xlabel("$x$", fontsize=12) plt.ylabel("$f(x)$", fontsize=12) plt.axhline(0, color='black', linewidth=0.8, linestyle='--') plt.axvline(0, color='black', linewidth=0.8, linestyle='--')

Highlight the steps explicitly

for i in range(-6, 3): plt.plot([i-2, i-1], [i, i], 'bo-', markersize=4)

plt.grid(True, linestyle='--', alpha=0.6) plt.legend() plt.show()The graph above shows the function $f(x) = [x+2]$, which is a step function with shifts to the left. The open and closed circles indicate the interval limits, where each step starts and ends.

Would you like a detailed explanation or additional modifications? Here are some related questions:

1. How is the floor function defined mathematically?
2. What are other transformations that can be applied to $f(x) = [x]$?
3. How would $f(x) = [x-2]$ differ in appearance?
4. Can we express the greatest integer function in terms of piecewise functions?
5. How does the floor function relate to other rounding functions?

Tip: Always look for open and closed intervals when interpreting step functions like the floor function.